Find the conformal maps from $\Omega =\left \{ z\in\mathbb{C}:|z|>1,Re(z)>0, \text{ and } Im(z)>0 \right \}$ to the unit disk

Question

Find all the conformal mappings from $\Omega$ to the unit disk,where
$\Omega=\{z\in\mathbb{C}: |z|>1, Re(z)>0, Im(z)>0 \}$.
Such that the image of $1+i$ is $0$.

I figured out that by taking the composition of
$T_1=iz+z$ (Or I guess $z^2$ will also work).
$T_2=\frac{1}{2}(z+\frac{1}{z})$
$T_3= \frac{i-z}{i+z}$
We can map the $\Omega$ to the unit disk.

But it doesn't satisfy the condition that $i+1\to 0$ .
Can I alter my answer to get the desired result or is there a totally a new approach.

Answer 1

To find one such mapping, first apply $z \mapsto z^2$. Then apply the Joukowsky transform $J$. Since $J(e^{i t}) = \cos t$, consider what the boundary of $\Omega$ is mapped to.